Consimilarity and quaternion matrix equations $AX-\hat{X}B=C$, $X-A\hat{X}B=C$

Abstract

L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix $A$ with respect to consimilarity transformations ${\tilde{S}}^{-1}AS$ in which $S$ is a nonsingular quaternion matrix and $\tilde{h}:=a-bi+cj-dk$ for each quaternion $h=a+bi+cj+dk$. We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations ${\hat{S}}^{-1}AS$ in which $h\mapsto \hat{h}$ is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$.

Cited by in corpus (1)

• Roth's solvability criteria for the matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$ over the skew field of quaternions with an involutive automorphism $q\mapsto \hat{q}$